Euler summation method
One of the methods for summing series of numbers and functions. A series
![]() | (*) |
is summable by means of the Euler summation method (-summable) to the sum
if
![]() |
where and
.
The method was first applied by L. Euler for to sum slowly-convergent or divergent series. Since the technique was later extended to arbitrary values of
by K. Knopp [1], it is also known for arbitrary
as the Euler–Knopp summation method. This method is regular for
(see Regular summation methods); if a series is
-summable, then it is also
-summable,
, to the same sum (see Inclusion of summation methods). For
the summability of the series (*) by the Euler summation method implies that the series is convergent. If the series is
-summable, then its terms
satisfy the condition
,
. The Euler summation method can also be applied for analytic continuation beyond the disc of convergence of a function defined by means of a power series. Thus, the series
is
-summable to the sum
in the disc with centre at
and of radius
.
References
[1] | K. Knopp, "Ueber das Eulersche Summierungsverfahren" Math. Z. , 15 (1922) pp. 226–253 |
[2] | K. Knopp, "Ueber das Eulersche Summierungsverfahren II" Math. Z. , 18 (1923) pp. 125–156 |
[3] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
[4] | S. Baron, "Introduction to theory of summation of series" , Tallin (1977) (In Russian) |
Comments
References
[a1] | K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970) |
Euler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_summation_method&oldid=32981